src/HOL/Orderings.thy
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     1 (*  Title:      HOL/Orderings.thy
       
     2     ID:         $Id$
       
     3     Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
       
     4 
       
     5 FIXME: derive more of the min/max laws generically via semilattices
       
     6 *)
       
     7 
       
     8 header {* Type classes for $\le$ *}
       
     9 
       
    10 theory Orderings
       
    11 imports Lattice_Locales
       
    12 files ("antisym_setup.ML")
       
    13 begin
       
    14 
       
    15 subsection {* Order signatures and orders *}
       
    16 
       
    17 axclass
       
    18   ord < type
       
    19 
       
    20 syntax
       
    21   "op <"        :: "['a::ord, 'a] => bool"             ("op <")
       
    22   "op <="       :: "['a::ord, 'a] => bool"             ("op <=")
       
    23 
       
    24 global
       
    25 
       
    26 consts
       
    27   "op <"        :: "['a::ord, 'a] => bool"             ("(_/ < _)"  [50, 51] 50)
       
    28   "op <="       :: "['a::ord, 'a] => bool"             ("(_/ <= _)" [50, 51] 50)
       
    29 
       
    30 local
       
    31 
       
    32 syntax (xsymbols)
       
    33   "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
       
    34   "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
       
    35 
       
    36 syntax (HTML output)
       
    37   "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
       
    38   "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
       
    39 
       
    40 text{* Syntactic sugar: *}
       
    41 
       
    42 consts
       
    43   "_gt" :: "'a::ord => 'a => bool"             (infixl ">" 50)
       
    44   "_ge" :: "'a::ord => 'a => bool"             (infixl ">=" 50)
       
    45 translations
       
    46   "x > y"  => "y < x"
       
    47   "x >= y" => "y <= x"
       
    48 
       
    49 syntax (xsymbols)
       
    50   "_ge"       :: "'a::ord => 'a => bool"             (infixl "\<ge>" 50)
       
    51 
       
    52 syntax (HTML output)
       
    53   "_ge"       :: "['a::ord, 'a] => bool"             (infixl "\<ge>" 50)
       
    54 
       
    55 
       
    56 subsection {* Monotonicity *}
       
    57 
       
    58 locale mono =
       
    59   fixes f
       
    60   assumes mono: "A <= B ==> f A <= f B"
       
    61 
       
    62 lemmas monoI [intro?] = mono.intro
       
    63   and monoD [dest?] = mono.mono
       
    64 
       
    65 constdefs
       
    66   min :: "['a::ord, 'a] => 'a"
       
    67   "min a b == (if a <= b then a else b)"
       
    68   max :: "['a::ord, 'a] => 'a"
       
    69   "max a b == (if a <= b then b else a)"
       
    70 
       
    71 lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
       
    72   by (simp add: min_def)
       
    73 
       
    74 lemma min_of_mono:
       
    75     "ALL x y. (f x <= f y) = (x <= y) ==> min (f m) (f n) = f (min m n)"
       
    76   by (simp add: min_def)
       
    77 
       
    78 lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
       
    79   by (simp add: max_def)
       
    80 
       
    81 lemma max_of_mono:
       
    82     "ALL x y. (f x <= f y) = (x <= y) ==> max (f m) (f n) = f (max m n)"
       
    83   by (simp add: max_def)
       
    84 
       
    85 
       
    86 subsection "Orders"
       
    87 
       
    88 axclass order < ord
       
    89   order_refl [iff]: "x <= x"
       
    90   order_trans: "x <= y ==> y <= z ==> x <= z"
       
    91   order_antisym: "x <= y ==> y <= x ==> x = y"
       
    92   order_less_le: "(x < y) = (x <= y & x ~= y)"
       
    93 
       
    94 text{* Connection to locale: *}
       
    95 
       
    96 lemma partial_order_order:
       
    97  "partial_order (op \<le> :: 'a::order \<Rightarrow> 'a \<Rightarrow> bool)"
       
    98 apply(rule partial_order.intro)
       
    99 apply(rule order_refl, erule (1) order_trans, erule (1) order_antisym)
       
   100 done
       
   101 
       
   102 text {* Reflexivity. *}
       
   103 
       
   104 lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y"
       
   105     -- {* This form is useful with the classical reasoner. *}
       
   106   apply (erule ssubst)
       
   107   apply (rule order_refl)
       
   108   done
       
   109 
       
   110 lemma order_less_irrefl [iff]: "~ x < (x::'a::order)"
       
   111   by (simp add: order_less_le)
       
   112 
       
   113 lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
       
   114     -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
       
   115   apply (simp add: order_less_le, blast)
       
   116   done
       
   117 
       
   118 lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
       
   119 
       
   120 lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
       
   121   by (simp add: order_less_le)
       
   122 
       
   123 
       
   124 text {* Asymmetry. *}
       
   125 
       
   126 lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"
       
   127   by (simp add: order_less_le order_antisym)
       
   128 
       
   129 lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P"
       
   130   apply (drule order_less_not_sym)
       
   131   apply (erule contrapos_np, simp)
       
   132   done
       
   133 
       
   134 lemma order_eq_iff: "!!x::'a::order. (x = y) = (x \<le> y & y \<le> x)"
       
   135 by (blast intro: order_antisym)
       
   136 
       
   137 lemma order_antisym_conv: "(y::'a::order) <= x ==> (x <= y) = (x = y)"
       
   138 by(blast intro:order_antisym)
       
   139 
       
   140 text {* Transitivity. *}
       
   141 
       
   142 lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z"
       
   143   apply (simp add: order_less_le)
       
   144   apply (blast intro: order_trans order_antisym)
       
   145   done
       
   146 
       
   147 lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
       
   148   apply (simp add: order_less_le)
       
   149   apply (blast intro: order_trans order_antisym)
       
   150   done
       
   151 
       
   152 lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
       
   153   apply (simp add: order_less_le)
       
   154   apply (blast intro: order_trans order_antisym)
       
   155   done
       
   156 
       
   157 
       
   158 text {* Useful for simplification, but too risky to include by default. *}
       
   159 
       
   160 lemma order_less_imp_not_less: "(x::'a::order) < y ==>  (~ y < x) = True"
       
   161   by (blast elim: order_less_asym)
       
   162 
       
   163 lemma order_less_imp_triv: "(x::'a::order) < y ==>  (y < x --> P) = True"
       
   164   by (blast elim: order_less_asym)
       
   165 
       
   166 lemma order_less_imp_not_eq: "(x::'a::order) < y ==>  (x = y) = False"
       
   167   by auto
       
   168 
       
   169 lemma order_less_imp_not_eq2: "(x::'a::order) < y ==>  (y = x) = False"
       
   170   by auto
       
   171 
       
   172 
       
   173 text {* Other operators. *}
       
   174 
       
   175 lemma min_leastR: "(!!x::'a::order. least <= x) ==> min x least = least"
       
   176   apply (simp add: min_def)
       
   177   apply (blast intro: order_antisym)
       
   178   done
       
   179 
       
   180 lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x"
       
   181   apply (simp add: max_def)
       
   182   apply (blast intro: order_antisym)
       
   183   done
       
   184 
       
   185 
       
   186 subsection {* Transitivity rules for calculational reasoning *}
       
   187 
       
   188 
       
   189 lemma order_neq_le_trans: "a ~= b ==> (a::'a::order) <= b ==> a < b"
       
   190   by (simp add: order_less_le)
       
   191 
       
   192 lemma order_le_neq_trans: "(a::'a::order) <= b ==> a ~= b ==> a < b"
       
   193   by (simp add: order_less_le)
       
   194 
       
   195 lemma order_less_asym': "(a::'a::order) < b ==> b < a ==> P"
       
   196   by (rule order_less_asym)
       
   197 
       
   198 
       
   199 subsection {* Least value operator *}
       
   200 
       
   201 constdefs
       
   202   Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
       
   203   "Least P == THE x. P x & (ALL y. P y --> x <= y)"
       
   204     -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
       
   205 
       
   206 lemma LeastI2:
       
   207   "[| P (x::'a::order);
       
   208       !!y. P y ==> x <= y;
       
   209       !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
       
   210    ==> Q (Least P)"
       
   211   apply (unfold Least_def)
       
   212   apply (rule theI2)
       
   213     apply (blast intro: order_antisym)+
       
   214   done
       
   215 
       
   216 lemma Least_equality:
       
   217     "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
       
   218   apply (simp add: Least_def)
       
   219   apply (rule the_equality)
       
   220   apply (auto intro!: order_antisym)
       
   221   done
       
   222 
       
   223 
       
   224 subsection "Linear / total orders"
       
   225 
       
   226 axclass linorder < order
       
   227   linorder_linear: "x <= y | y <= x"
       
   228 
       
   229 lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x"
       
   230   apply (simp add: order_less_le)
       
   231   apply (insert linorder_linear, blast)
       
   232   done
       
   233 
       
   234 lemma linorder_le_less_linear: "!!x::'a::linorder. x\<le>y | y<x"
       
   235   by (simp add: order_le_less linorder_less_linear)
       
   236 
       
   237 lemma linorder_le_cases [case_names le ge]:
       
   238     "((x::'a::linorder) \<le> y ==> P) ==> (y \<le> x ==> P) ==> P"
       
   239   by (insert linorder_linear, blast)
       
   240 
       
   241 lemma linorder_cases [case_names less equal greater]:
       
   242     "((x::'a::linorder) < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P"
       
   243   by (insert linorder_less_linear, blast)
       
   244 
       
   245 lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)"
       
   246   apply (simp add: order_less_le)
       
   247   apply (insert linorder_linear)
       
   248   apply (blast intro: order_antisym)
       
   249   done
       
   250 
       
   251 lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
       
   252   apply (simp add: order_less_le)
       
   253   apply (insert linorder_linear)
       
   254   apply (blast intro: order_antisym)
       
   255   done
       
   256 
       
   257 lemma linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)"
       
   258 by (cut_tac x = x and y = y in linorder_less_linear, auto)
       
   259 
       
   260 lemma linorder_neqE: "x ~= (y::'a::linorder) ==> (x < y ==> R) ==> (y < x ==> R) ==> R"
       
   261 by (simp add: linorder_neq_iff, blast)
       
   262 
       
   263 lemma linorder_antisym_conv1: "~ (x::'a::linorder) < y ==> (x <= y) = (x = y)"
       
   264 by(blast intro:order_antisym dest:linorder_not_less[THEN iffD1])
       
   265 
       
   266 lemma linorder_antisym_conv2: "(x::'a::linorder) <= y ==> (~ x < y) = (x = y)"
       
   267 by(blast intro:order_antisym dest:linorder_not_less[THEN iffD1])
       
   268 
       
   269 lemma linorder_antisym_conv3: "~ (y::'a::linorder) < x ==> (~ x < y) = (x = y)"
       
   270 by(blast intro:order_antisym dest:linorder_not_less[THEN iffD1])
       
   271 
       
   272 use "antisym_setup.ML";
       
   273 setup antisym_setup
       
   274 
       
   275 subsection {* Setup of transitivity reasoner as Solver *}
       
   276 
       
   277 lemma less_imp_neq: "[| (x::'a::order) < y |] ==> x ~= y"
       
   278   by (erule contrapos_pn, erule subst, rule order_less_irrefl)
       
   279 
       
   280 lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
       
   281   by (erule subst, erule ssubst, assumption)
       
   282 
       
   283 ML_setup {*
       
   284 
       
   285 (* The setting up of Quasi_Tac serves as a demo.  Since there is no
       
   286    class for quasi orders, the tactics Quasi_Tac.trans_tac and
       
   287    Quasi_Tac.quasi_tac are not of much use. *)
       
   288 
       
   289 fun decomp_gen sort sign (Trueprop $ t) =
       
   290   let fun of_sort t = Sign.of_sort sign (type_of t, sort)
       
   291   fun dec (Const ("Not", _) $ t) = (
       
   292 	  case dec t of
       
   293 	    None => None
       
   294 	  | Some (t1, rel, t2) => Some (t1, "~" ^ rel, t2))
       
   295 	| dec (Const ("op =",  _) $ t1 $ t2) =
       
   296 	    if of_sort t1
       
   297 	    then Some (t1, "=", t2)
       
   298 	    else None
       
   299 	| dec (Const ("op <=",  _) $ t1 $ t2) =
       
   300 	    if of_sort t1
       
   301 	    then Some (t1, "<=", t2)
       
   302 	    else None
       
   303 	| dec (Const ("op <",  _) $ t1 $ t2) =
       
   304 	    if of_sort t1
       
   305 	    then Some (t1, "<", t2)
       
   306 	    else None
       
   307 	| dec _ = None
       
   308   in dec t end;
       
   309 
       
   310 structure Quasi_Tac = Quasi_Tac_Fun (
       
   311   struct
       
   312     val le_trans = thm "order_trans";
       
   313     val le_refl = thm "order_refl";
       
   314     val eqD1 = thm "order_eq_refl";
       
   315     val eqD2 = thm "sym" RS thm "order_eq_refl";
       
   316     val less_reflE = thm "order_less_irrefl" RS thm "notE";
       
   317     val less_imp_le = thm "order_less_imp_le";
       
   318     val le_neq_trans = thm "order_le_neq_trans";
       
   319     val neq_le_trans = thm "order_neq_le_trans";
       
   320     val less_imp_neq = thm "less_imp_neq";
       
   321     val decomp_trans = decomp_gen ["Orderings.order"];
       
   322     val decomp_quasi = decomp_gen ["Orderings.order"];
       
   323 
       
   324   end);  (* struct *)
       
   325 
       
   326 structure Order_Tac = Order_Tac_Fun (
       
   327   struct
       
   328     val less_reflE = thm "order_less_irrefl" RS thm "notE";
       
   329     val le_refl = thm "order_refl";
       
   330     val less_imp_le = thm "order_less_imp_le";
       
   331     val not_lessI = thm "linorder_not_less" RS thm "iffD2";
       
   332     val not_leI = thm "linorder_not_le" RS thm "iffD2";
       
   333     val not_lessD = thm "linorder_not_less" RS thm "iffD1";
       
   334     val not_leD = thm "linorder_not_le" RS thm "iffD1";
       
   335     val eqI = thm "order_antisym";
       
   336     val eqD1 = thm "order_eq_refl";
       
   337     val eqD2 = thm "sym" RS thm "order_eq_refl";
       
   338     val less_trans = thm "order_less_trans";
       
   339     val less_le_trans = thm "order_less_le_trans";
       
   340     val le_less_trans = thm "order_le_less_trans";
       
   341     val le_trans = thm "order_trans";
       
   342     val le_neq_trans = thm "order_le_neq_trans";
       
   343     val neq_le_trans = thm "order_neq_le_trans";
       
   344     val less_imp_neq = thm "less_imp_neq";
       
   345     val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq";
       
   346     val decomp_part = decomp_gen ["Orderings.order"];
       
   347     val decomp_lin = decomp_gen ["Orderings.linorder"];
       
   348 
       
   349   end);  (* struct *)
       
   350 
       
   351 simpset_ref() := simpset ()
       
   352     addSolver (mk_solver "Trans_linear" (fn _ => Order_Tac.linear_tac))
       
   353     addSolver (mk_solver "Trans_partial" (fn _ => Order_Tac.partial_tac));
       
   354   (* Adding the transitivity reasoners also as safe solvers showed a slight
       
   355      speed up, but the reasoning strength appears to be not higher (at least
       
   356      no breaking of additional proofs in the entire HOL distribution, as
       
   357      of 5 March 2004, was observed). *)
       
   358 *}
       
   359 
       
   360 (* Optional setup of methods *)
       
   361 
       
   362 (*
       
   363 method_setup trans_partial =
       
   364   {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (Order_Tac.partial_tac)) *}
       
   365   {* transitivity reasoner for partial orders *}	
       
   366 method_setup trans_linear =
       
   367   {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (Order_Tac.linear_tac)) *}
       
   368   {* transitivity reasoner for linear orders *}
       
   369 *)
       
   370 
       
   371 (*
       
   372 declare order.order_refl [simp del] order_less_irrefl [simp del]
       
   373 
       
   374 can currently not be removed, abel_cancel relies on it.
       
   375 *)
       
   376 
       
   377 
       
   378 subsection "Min and max on (linear) orders"
       
   379 
       
   380 lemma min_same [simp]: "min (x::'a::order) x = x"
       
   381   by (simp add: min_def)
       
   382 
       
   383 lemma max_same [simp]: "max (x::'a::order) x = x"
       
   384   by (simp add: max_def)
       
   385 
       
   386 text{* Instantiate locales: *}
       
   387 
       
   388 lemma lower_semilattice_lin_min:
       
   389   "lower_semilattice(op \<le>) (min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
       
   390 apply(rule lower_semilattice.intro)
       
   391 apply(rule partial_order_order)
       
   392 apply(rule lower_semilattice_axioms.intro)
       
   393 apply(simp add:min_def linorder_not_le order_less_imp_le)
       
   394 apply(simp add:min_def linorder_not_le order_less_imp_le)
       
   395 apply(simp add:min_def linorder_not_le order_less_imp_le)
       
   396 done
       
   397 
       
   398 lemma upper_semilattice_lin_max:
       
   399   "upper_semilattice(op \<le>) (max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
       
   400 apply(rule upper_semilattice.intro)
       
   401 apply(rule partial_order_order)
       
   402 apply(rule upper_semilattice_axioms.intro)
       
   403 apply(simp add: max_def linorder_not_le order_less_imp_le)
       
   404 apply(simp add: max_def linorder_not_le order_less_imp_le)
       
   405 apply(simp add: max_def linorder_not_le order_less_imp_le)
       
   406 done
       
   407 
       
   408 lemma lattice_min_max: "lattice (op \<le>) (min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a) max"
       
   409 apply(rule lattice.intro)
       
   410 apply(rule partial_order_order)
       
   411 apply(rule lower_semilattice.axioms[OF lower_semilattice_lin_min])
       
   412 apply(rule upper_semilattice.axioms[OF upper_semilattice_lin_max])
       
   413 done
       
   414 
       
   415 lemma distrib_lattice_min_max:
       
   416  "distrib_lattice (op \<le>) (min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a) max"
       
   417 apply(rule distrib_lattice.intro)
       
   418 apply(rule partial_order_order)
       
   419 apply(rule lower_semilattice.axioms[OF lower_semilattice_lin_min])
       
   420 apply(rule upper_semilattice.axioms[OF upper_semilattice_lin_max])
       
   421 apply(rule distrib_lattice_axioms.intro)
       
   422 apply(rule_tac x=x and y=y in linorder_le_cases)
       
   423 apply(rule_tac x=x and y=z in linorder_le_cases)
       
   424 apply(rule_tac x=y and y=z in linorder_le_cases)
       
   425 apply(simp add:min_def max_def)
       
   426 apply(simp add:min_def max_def)
       
   427 apply(rule_tac x=y and y=z in linorder_le_cases)
       
   428 apply(simp add:min_def max_def)
       
   429 apply(simp add:min_def max_def)
       
   430 apply(rule_tac x=x and y=z in linorder_le_cases)
       
   431 apply(rule_tac x=y and y=z in linorder_le_cases)
       
   432 apply(simp add:min_def max_def)
       
   433 apply(simp add:min_def max_def)
       
   434 apply(rule_tac x=y and y=z in linorder_le_cases)
       
   435 apply(simp add:min_def max_def)
       
   436 apply(simp add:min_def max_def)
       
   437 done
       
   438 
       
   439 lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
       
   440   apply(simp add:max_def)
       
   441   apply (insert linorder_linear)
       
   442   apply (blast intro: order_trans)
       
   443   done
       
   444 
       
   445 lemma le_maxI1: "(x::'a::linorder) <= max x y"
       
   446 by(rule upper_semilattice.sup_ge1[OF upper_semilattice_lin_max])
       
   447 
       
   448 lemma le_maxI2: "(y::'a::linorder) <= max x y"
       
   449     -- {* CANNOT use with @{text "[intro!]"} because blast will give PROOF FAILED. *}
       
   450 by(rule upper_semilattice.sup_ge2[OF upper_semilattice_lin_max])
       
   451 
       
   452 lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
       
   453   apply (simp add: max_def order_le_less)
       
   454   apply (insert linorder_less_linear)
       
   455   apply (blast intro: order_less_trans)
       
   456   done
       
   457 
       
   458 lemma max_le_iff_conj [simp]:
       
   459     "!!z::'a::linorder. (max x y <= z) = (x <= z & y <= z)"
       
   460 by (rule upper_semilattice.above_sup_conv[OF upper_semilattice_lin_max])
       
   461 
       
   462 lemma max_less_iff_conj [simp]:
       
   463     "!!z::'a::linorder. (max x y < z) = (x < z & y < z)"
       
   464   apply (simp add: order_le_less max_def)
       
   465   apply (insert linorder_less_linear)
       
   466   apply (blast intro: order_less_trans)
       
   467   done
       
   468 
       
   469 lemma le_min_iff_conj [simp]:
       
   470     "!!z::'a::linorder. (z <= min x y) = (z <= x & z <= y)"
       
   471     -- {* @{text "[iff]"} screws up a @{text blast} in MiniML *}
       
   472 by (rule lower_semilattice.below_inf_conv[OF lower_semilattice_lin_min])
       
   473 
       
   474 lemma min_less_iff_conj [simp]:
       
   475     "!!z::'a::linorder. (z < min x y) = (z < x & z < y)"
       
   476   apply (simp add: order_le_less min_def)
       
   477   apply (insert linorder_less_linear)
       
   478   apply (blast intro: order_less_trans)
       
   479   done
       
   480 
       
   481 lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)"
       
   482   apply (simp add: min_def)
       
   483   apply (insert linorder_linear)
       
   484   apply (blast intro: order_trans)
       
   485   done
       
   486 
       
   487 lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)"
       
   488   apply (simp add: min_def order_le_less)
       
   489   apply (insert linorder_less_linear)
       
   490   apply (blast intro: order_less_trans)
       
   491   done
       
   492 
       
   493 lemma max_assoc: "!!x::'a::linorder. max (max x y) z = max x (max y z)"
       
   494 by (rule upper_semilattice.sup_assoc[OF upper_semilattice_lin_max])
       
   495 
       
   496 lemma max_commute: "!!x::'a::linorder. max x y = max y x"
       
   497 by (rule upper_semilattice.sup_commute[OF upper_semilattice_lin_max])
       
   498 
       
   499 lemmas max_ac = max_assoc max_commute
       
   500                 mk_left_commute[of max,OF max_assoc max_commute]
       
   501 
       
   502 lemma min_assoc: "!!x::'a::linorder. min (min x y) z = min x (min y z)"
       
   503 by (rule lower_semilattice.inf_assoc[OF lower_semilattice_lin_min])
       
   504 
       
   505 lemma min_commute: "!!x::'a::linorder. min x y = min y x"
       
   506 by (rule lower_semilattice.inf_commute[OF lower_semilattice_lin_min])
       
   507 
       
   508 lemmas min_ac = min_assoc min_commute
       
   509                 mk_left_commute[of min,OF min_assoc min_commute]
       
   510 
       
   511 lemma split_min:
       
   512     "P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"
       
   513   by (simp add: min_def)
       
   514 
       
   515 lemma split_max:
       
   516     "P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"
       
   517   by (simp add: max_def)
       
   518 
       
   519 
       
   520 subsection "Bounded quantifiers"
       
   521 
       
   522 syntax
       
   523   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3ALL _<_./ _)"  [0, 0, 10] 10)
       
   524   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3EX _<_./ _)"  [0, 0, 10] 10)
       
   525   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3ALL _<=_./ _)" [0, 0, 10] 10)
       
   526   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3EX _<=_./ _)" [0, 0, 10] 10)
       
   527 
       
   528   "_gtAll" :: "[idt, 'a, bool] => bool"   ("(3ALL _>_./ _)"  [0, 0, 10] 10)
       
   529   "_gtEx"  :: "[idt, 'a, bool] => bool"   ("(3EX _>_./ _)"  [0, 0, 10] 10)
       
   530   "_geAll"   :: "[idt, 'a, bool] => bool"   ("(3ALL _>=_./ _)" [0, 0, 10] 10)
       
   531   "_geEx"    :: "[idt, 'a, bool] => bool"   ("(3EX _>=_./ _)" [0, 0, 10] 10)
       
   532 
       
   533 syntax (xsymbols)
       
   534   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
       
   535   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
       
   536   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
       
   537   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
       
   538 
       
   539   "_gtAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
       
   540   "_gtEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
       
   541   "_geAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
       
   542   "_geEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
       
   543 
       
   544 syntax (HOL)
       
   545   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
       
   546   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
       
   547   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
       
   548   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
       
   549 
       
   550 syntax (HTML output)
       
   551   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
       
   552   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
       
   553   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
       
   554   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
       
   555 
       
   556   "_gtAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
       
   557   "_gtEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
       
   558   "_geAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
       
   559   "_geEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
       
   560 
       
   561 translations
       
   562  "ALL x<y. P"   =>  "ALL x. x < y --> P"
       
   563  "EX x<y. P"    =>  "EX x. x < y  & P"
       
   564  "ALL x<=y. P"  =>  "ALL x. x <= y --> P"
       
   565  "EX x<=y. P"   =>  "EX x. x <= y & P"
       
   566  "ALL x>y. P"   =>  "ALL x. x > y --> P"
       
   567  "EX x>y. P"    =>  "EX x. x > y  & P"
       
   568  "ALL x>=y. P"  =>  "ALL x. x >= y --> P"
       
   569  "EX x>=y. P"   =>  "EX x. x >= y & P"
       
   570 
       
   571 print_translation {*
       
   572 let
       
   573   fun mk v v' q n P =
       
   574     if v=v' andalso not(v  mem (map fst (Term.add_frees([],n))))
       
   575     then Syntax.const q $ Syntax.mark_bound v' $ n $ P else raise Match;
       
   576   fun all_tr' [Const ("_bound",_) $ Free (v,_),
       
   577                Const("op -->",_) $ (Const ("op <",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] =
       
   578     mk v v' "_lessAll" n P
       
   579 
       
   580   | all_tr' [Const ("_bound",_) $ Free (v,_),
       
   581                Const("op -->",_) $ (Const ("op <=",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] =
       
   582     mk v v' "_leAll" n P
       
   583 
       
   584   | all_tr' [Const ("_bound",_) $ Free (v,_),
       
   585                Const("op -->",_) $ (Const ("op <",_) $ n $ (Const ("_bound",_) $ Free (v',_))) $ P] =
       
   586     mk v v' "_gtAll" n P
       
   587 
       
   588   | all_tr' [Const ("_bound",_) $ Free (v,_),
       
   589                Const("op -->",_) $ (Const ("op <=",_) $ n $ (Const ("_bound",_) $ Free (v',_))) $ P] =
       
   590     mk v v' "_geAll" n P;
       
   591 
       
   592   fun ex_tr' [Const ("_bound",_) $ Free (v,_),
       
   593                Const("op &",_) $ (Const ("op <",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] =
       
   594     mk v v' "_lessEx" n P
       
   595 
       
   596   | ex_tr' [Const ("_bound",_) $ Free (v,_),
       
   597                Const("op &",_) $ (Const ("op <=",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] =
       
   598     mk v v' "_leEx" n P
       
   599 
       
   600   | ex_tr' [Const ("_bound",_) $ Free (v,_),
       
   601                Const("op &",_) $ (Const ("op <",_) $ n $ (Const ("_bound",_) $ Free (v',_))) $ P] =
       
   602     mk v v' "_gtEx" n P
       
   603 
       
   604   | ex_tr' [Const ("_bound",_) $ Free (v,_),
       
   605                Const("op &",_) $ (Const ("op <=",_) $ n $ (Const ("_bound",_) $ Free (v',_))) $ P] =
       
   606     mk v v' "_geEx" n P
       
   607 in
       
   608 [("ALL ", all_tr'), ("EX ", ex_tr')]
       
   609 end
       
   610 *}
       
   611 
       
   612 end